NILMANIFOLDS
YVES CORNULIER
Abstract. Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole.
In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms of discrete cocompact subrings of the Lie algebra, making it more practical to estimate.
After providing some general upper bounds, we develop methods to provide nontrivial lower bounds. We provide the first computations of the asymptotics of the systolic growth of nilpotent groups for which this is not equivalent to the volume growth. In particular, we provide an example for which the degree of growth is not an integer; it has dimension 7. Finally, we gather some open questions.
1. Introduction
1.1. Background. Every locally compact groupGhas a Haar measureµ, unique up to positive scalar multiplication. If in additionGis generated by a symmetric compact neighborhood S of 1, the function b(n) = µ(Sn) is called the volume growth (or word growth) of G ; while its values depend on the choice of (S, µ), its asymptotics (in the usual meaning, recalled in §3.1) does not. The volume growth is either exponential or subexponential. Those compactly generated lo- cally compact group with polynomially bounded growth have been characterized by Guivarch and Jenkins [Gui, Jen] in the case of connected Lie groups, Gro- mov in the case of discrete groups [Gro1], and Losert [Los] in general. All such groups are commable, and hence quasi-isometric, to simply connected nilpotent Lie groups, and thus, by work of Guivarch [Gui] have an integral degree of poly- nomial growth that is easily computable in terms of the Lie algebra structure (see §2.1).
The object of study of the paper is the following related notion of growth, introduced by Gromov in [Gro2, p. 333].
Date: August 5, 2017.
2010Mathematics Subject Classification. Primary 17B30; Secondary 20F18, 20F69, 20E07, 22E25, 22E40, 53C23, 53C30.
This material is based upon work supported by the NSF under Grant No. DMS-1440140 while the author was in residence at the MSRI (Berkeley) during the Fall 2016 semester. Supported by ANR Project ANR-14-CE25-0004 GAMME.
1
Definition 1.1. Let H be a locally compact group and | · | the word length relative to some choice of compact generating subset. If X ⊂ H, define its systole as sys(X) = inf{|x|:x∈Xr{1}} ∈R+∪ {+∞}.
EndowHwith some left-invariant Haar measure. The systolic growth of (H,|·|) is the function mapping r ≥ 0 to the infimumσ(r) ∈R+∪ {+∞} of covolumes of cocompact lattices of H with systole ≥r.
See Remark 3.1 for the geometric interpretation in a Riemannian setting. Note that the definition makes sense when H is discrete, in which case lattices just refer to finite index subgroups: this is actually the setting in Gromov’s original definition. In the setting we will focus on,H will always be nilpotent and in this case all lattices are cocompact. In general, we can define another type of systolic growth, allowing non-cocompact lattices.
The asymptotics of the growth of σ does not depend on the choice of the word length. The number σ(r) is always bounded below by the volume of the open r/2-ball in H.
The functionσ is interesting only when it takes finite values, in which case we say that H is residually systolic. When H is discrete, this just means that H is residually finite. In general, a sufficient condition for H being residually systolic is that H admits a residually finite cocompact lattice.
It is natural to compare the volume growth and the systolic growth. For finitely generated linear groups of exponential growth, the systolic growth is exponential as well [BouCor].
1.2. Background with focus in the nilpotent case. Given a Lie algebra g, denote by (gi)i≥1 its lower central series (see §2.1); by definition g is c-step nilpotent if gc+1 = {0}. The homogeneous dimension of g is classically defined as the sum
D=D(g) = X
i≥1
dim(gi);
we have D <∞ if and only g is nilpotent and finite-dimensional.
A classical result of Malcev is that a simply connected nilpotent Lie group ad- mits a lattice (which is then cocompact and residually finite) precisely when its Lie algebra can be obtained from a rational Lie algebra by extension of scalars.
Therefore this is also equivalent to being residually systolic. In this case, the sys- tolic growth is easy to bound polynomially; nevertheless the comparison between volume growth and the systolic growth is not obvious, because the precise rate of polynomial growth is an issue.
A first step towards a good understanding is the following result (all asymptotic results are meant when r →+∞).
Theorem 1.2 ([Cor]). Let G be a simply connected nilpotent Lie group with a lattice Γ. Let g be the Lie algebra of G, and let D be its homogeneous dimension
(so, for both G and Γ, the growth is ' r 7→ rD and the systolic growth is rD, see §2.1). The following are equivalent:
(1) the systolic growth of G is 'rD; (2) the systolic growth of Γ is 'rD;
(3) g is Carnot, i.e., admits a Lie algebra grading g = L
i≥1gi such that g1 generates g.
Otherwise, if σ denotes the systolic growth of either Γ or G, then it satisfies σ(r)rD.
In the non-Carnot case, the proof of Theorem 1.2 does not provide any explicit asymptotic lower bound improving σ(r) rD. In this paper, we carry out the task of evaluating the systolic growth in a number of explicit non-Carnot cases.
Such results are presented in§1.4. We start with some general upper bounds. We will often emphasize the quotientσ0(r) =r−Dσ(r), since it often naturally occurs in computations, and its growth is a measure of the failure of being Carnot.
1.3. Upper bounds. We provide here some upper bounds on the systolic growth.
We denote by d·e the ceiling function. Given c≥0, we define kc(g) =
dc/2e−1
X
i=1
c 2 −i
dim(gi/gi+1);
Note thatkc(g)≤ c2 −1
dim(g/gdc/2e).
Proposition 1.3(See Proposition 5.1). Letgbe a finite-dimensionalc-step nilpo- tent real Lie algebra with homogeneous dimensionD, and let k=kc(g) be defined as above. Assume that the corresponding simply connected nilpotent Lie groupG admits lattices. Then the systolic growth σ(r) of G and its lattices is rD+k.
We havekc(g)≤ 16dim(g)2(see Proposition 5.2) whencis the nilpotency length of g, so that we obtain an upper bound on σ(r)/rD depending only on dim(g).
For small values ofc, we have k≤2(g) = 0; k3(g) = 1
2dim(g/g2), k4(g) = dim(g/g2);
k5(g) = 3
2dim(g/g2) + 1
2dim(g2/g3); k6(g) = 2 dim(g/g2) + dim(g2/g3).
Also note thatD(g) +kc(g)≤cdim(g)/2. This improves the trivial upper bound rcdim(g), making use of congruence subgroups in a lattice, which was mentioned in [Cor].
Note that every 2-step nilpotent Lie algebra is Carnot; the smallest nilpotency length allowing non-Carnot Lie algebra is 3. When g is 3-step nilpotent, the above proposition yields σ(r) rD+dim(g/g2)/2. This bound is not very far from sharp, see Theorem 1.7.
1.4. Lower bounds on the systolic growth and precise estimates. This part is the bulk of the paper. It contains the first exact estimates of the asymp- totic behavior of the systolic growth of nilpotent Lie groups beyond the Carnot case covered by Theorem 1.2. The first non-Carnot Lie algebras occur in dimen- sion 5 and in this case we have the following theorem.
Theorem 1.4. Let G be a 5-dimensional simply connected nilpotent Lie group whose Lie algebra g is non-Carnot (there are 2 non-isomorphic possibilities for g, for which the homogeneous dimension D is either 8 or 11). Then the systolic growth of both G and its lattices is 'rD+1.
Both cases are obtained in a single proof. In dimension 6, the classification yields 13 non-Carnot nilpotent real Lie algebras; a similar approach provides precise estimates for at least some of them, but I do not know if it can exhibit a behavior different from being 'rD+h withh∈ {1,2,3}. In dimension 7, where a classification is still known (but lengthy), a similar approach yields an example for which the degree is not an integer:
Theorem 1.5. There exists a 7-dimensional simply connected nilpotent Lie group for which the systolic growth, as well as the systolic growth of one of its lattices, is 'rD+3/2.
This contrasts with the fact that the volume growth always has an integral degree of polynomial growth (the homogeneous dimensionD). Yet so far we only know, for the systolic growth, behaviors of the form rD+h with h a non-negative rational, but we actually do not know if logσ(r)/log(r) always converges, and if so, if its limit is always a rational, and what kind of further constraints we can expect on h (see the questions below).
At the computational level, let us also provide some families of unbounded dimension, for which we obtain unbounded values for h.
Theorem 1.6 (Truncated Witt Lie algebra). For n ≥3, let G(n) be the simply connected nilpotent Lie group corresponding to the Lie algebra g(n) with basis (ei)1≤i≤n and nonzero brackets[ei, ej] = (i−j)ei+j, (i+j ≤n). Then its systolic growth grows asr7→rD+h withh=d(n−4)/2e(here the homogeneous dimension is D= n(n−1)2 + 1).
The following family of examples with unboundedhconsists of 3-step nilpotent Lie groups.
Theorem 1.7 (see Theorem 7.6). For n ≥0, letg(4 + 2n) be the 3-step nilpotent (4 + 2n)-dimensional Lie algebra obtained as central product of a 4-dimensional filiform Lie algebra and a(2n+ 1)-dimensional Heisenberg Lie algebra, andG(4 + 2n) the corresponding simply connected nilpotent Lie group. Then its systolic growth grows as r7→rD+n, where D= 2n+ 7 is the homogeneous dimension.
The same method actually yields examples for which the polynomial degree of r−Dσ(r) is comparable to the square of the dimension, see Remark 7.8.
1.5. Outline of the method. Let us outline the method used to obtained the estimates of §1.4.
The first step is to translate the problem, which concerns lattices in a simply connected nilpotent Lie group, into a problem about discrete cocompact subrings in its Lie algebra. This uses the fact that even if there is no exact correspondence between the two (the exponential of a Lie subring can fail to be a subgroup, and the logarithm of a subgroup can fail to be a Lie subring), this is true “up to bounded index”. This important fact, for which we claim no originality but could not refer to a written proof, is formulated in Lemma 4.1 and proved (along with a more precise statement) in the appendix.
Then the notion of systolic growth can be made meaningful in a real finite- dimensional nilpotent Lie algebra: it mapsrto the smallest covolume of a discrete cocompact subring of Guivarch systole ≥ r. Here the Guivarch systole is the Lie algebra counterpart of the systole: this is the smallest Guivarch length of a nontrivial element in the lattice. The Guivarch length is recalled in §2.1; for instance, in the 3-dimensional Heisenberg Lie algebra, the Guivarch length of an element
0 x z 0 0 y 0 0 0
can be defined as the value |x|+|y|+|z|1/2. See§4.2. The previous fact shows that the systolic growth of a simply connected nilpotent Lie group is asymptotically equivalent to that of its Lie algebra.
Next, we have to estimate the systolic growth in various Lie algebras. The idea is to use a flag of rational ideals g = w1 ≥ . . . .wk = {0}. Here, if we consider arbitrary lattices, we need these ideals to be rational for every rational structure (we call this solid and provide some basic fact about such ideals in
§2.2). For instance, terms of the lower central series are such ideals. Then any lattice intersects eachwi into a lattice and this intersection maps into a lattice in wi/wi+1; let ai be the corresponding covolume. Then the covolume of the whole lattice is Q
iai. Then we use the stability under brackets and the hypothesis of Guivarch systole ≥ r to obtain lower bounds on Q
ai, which in some cases are better than the trivial lower bound (the trivial lower bound has the form rD).
More precisely, this approach typically yields, for a lattice of Guivarch systole
≥ r, some inequalities of the form aiaj ≥ akrm(i,j,k) for some integer (i, j, k). If we write Ai = logr(ai) (so that the covolume is rPAi), this can be rewritten as Ai+Aj ≥ Ak+m(i, j, k). Then such a family of inequalities can yield a lower bound of the formP
Ai ≥q for some rationalq, and thus yielding a lower bound for the covolumeQ
ai ≥rq.
Once such a method is checked to yield precise estimates in some cases, it is not a surprise to find that in well-chosen examples, it yields non-integral degrees, as in Theorem 1.5.
Let us also mention that we actually renormalize the problem by a well-chosen family of linear automorphisms (§6.1), which yields lower bounds for r−Dσ(r)
and simplifies the computations (for instance, it allows to treat simultaneously both examples of Theorem 1.4). The approach also provides a new, simpler proof of the implication (1)⇒(3) in Theorem 1.2, see§6.3.
1.6. Open questions. Let us mention some open problems. Since the numberh is possibly not always defined, we write things as follows. LetH be a compactly generated, locally compact group of polynomial growth, admitting at least a lattice (we especially have in mind the cases when H is a simply connected nilpotent Lie group, or H is a finitely generated nilpotent group). So its systolic growth r 7→σ(r) is well-defined. Let D be its homogeneous dimension. Define
hH = limlog(σ(n)/nD)
log(n) ; hH = limlog(σ(n)/nD) log(n) Question 1.8.
(1) Is it always true that hH =hH? (I conjecture a positive answer).
(2) ArehH and hH always rational numbers?
(3) Is it true that σH(n)' nD+h0 for some h0 ≥ 0? (Of course this implies a positive answer to (1), but this is more optimistic and I do not conjecture anything.)
Question 1.9. Does there exist infinitely many non-equivalent types of systolic growth asymptotically bounded above by some given polynomial?
Question 1.10. Let G be a simply connected nilpotent Lie group with a lattice Γ, with systolic growth σG and σΓ (so σGσΓ).
(1) Do we always haveσG 'σΓ?
(2) We now refer to the uniform systolic growth introduced in§3.3. We have σG σGu σΓ,Gu . Do we always have σG ' σuG? Do we always have σuG'σΓ,Gu ?
Acknowledgements. I thank Yves Benoist for useful hints, and Pierre de la Harpe for corrections on a preliminary version of this paper. I thank the referee for various corrections and useful references.
Contents
1. Introduction 1
2. Algebraic preliminaries 7
3. Systolic growths: facts and bounds 10
4. Algebraization of the systolic growth 13
5. General upper bounds on the systolic growth 14
6. The strategy for lower bounds 17
7. Precise estimates 19
Appendix A. Lattices and discrete cocompact subrings: back and forth 27
References 30
2. Algebraic preliminaries
In all this section,K is a field of characteristic zero, unless explicitly specified.
2.1. Lie algebras: lower central series, growth. Let g be a nilpotent fd Lie algebra over K (fd stands for finite-dimensional). Its lower central series is defined by g1 = g, gi+1 = [g,gi] for i ≥ 1. Let vi be a supplement subspace of gi+1 ingi, so that g=L
i≥1vi (and vi = 0 for large i). We have, for all i, j [vi,vj]⊂ M
k≥i+j
vk, vi+1 ⊂[v1,vi] +gi+2.
LetGbe the group ofK-points of the corresponding unipotent algebraic group;
Gcan obtained fromgusing the Baker-Campbell-Hausdorff formula as group law.
When K =R, this is the simply connected Lie group associated to g.
The integer
D=D(g) =D(G) = X
idim(vi) =X
i
dimgi
is called the homogeneous dimension ofG. Indeed, in the real case, the volume of ther-ball is ' rD [Gui] (see§3.1 for the definition of '); this degree formula was also found by Bass [Bas] while restricting to the discrete setting. We have D≥dim(g), with equality if and only if g is abelian.
Again in the real case, fix a norm on each vi. If x = (x1, x2, . . .) ∈ g in the decomposition g = L
i≥1vi, define its Guivarch length bxc = supkxik1/i. This length plays an important role, as Guivarch established that bxc is a good estimate for the word length of exp(x) in the simply connected nilpotent Lie groupG associated to g.
2.2. Solid ideals. We introduce here the notion of solid ideals, which will be useful when computing lower bounds on the systolic growth of simply connected nilpotent Lie groups.
Letgbe a Lie algebra (overK). AQ-structure ongis the data of aQ-subspace lsuch that the canonical homomorphism j :l⊗QK →g is a linear isomorphism of LieK-algebras. Ifl is a Q-subalgebra, we call it a multiplicative Q-structure, and then j is a K-algebra isomorphism.
Given aQ-structurel, aK-subspaceV ofgis calledQ-defined if it is generated as aK-subspace byV ∩l. If g is a LieK-algebra, we say that an ideal is solid if it isQ-defined for every multiplicativeQ-structure. The following properties are straightforward.
• A ideal contained and solid in a solid ideal is solid in the whole Lie algebra;
• the inverse image of a solid ideal by the quotient by a solid ideal is solid;
• the bracket of two solid ideals is solid:
• the centralizer of a solid ideal is solid;
• the intersection and the sum of two solid subalgebras are solid.
For instance, if g is abelian, then the only solid ideals of g are {0} and g.
To single out solid ideals in fdQ-definable real nilpotent Lie algebrasgis useful in view of the following: an ideal V ⊂ g is solid if and only if for every discrete cocompact subring Λ, the intersection Λ∩V is a lattice inV (or equivalently, the projection ong/V is a lattice ing/V). The reason is that theQ-linear span of any discrete cocompact subring is a multiplicative Q-structure, and that conversely any multiplicative Q-structure contains a discrete cocompact subring.
Solid ideals can almost be recognized using how they behave under the auto- morphism group. Say that an ideal I ingis absolutely Aut-invariant if, denoting by ¯K an algebraic closure ofK and Aut(g)K¯ for the group of automorphisms of the ¯K-algebra g⊗KK, the ideal¯ I ⊗K K¯ is Aut(g)K¯-invariant.
Theorem 2.1. Assume that K is uncountable. Let gbe a fd Lie algebra over K.
Then
(1) every solid ideal is invariant under Aut(g)0 (or equivalently, stable under all K-linear self-derivations of g);
(2) every absolutely Aut-invariant ideal is solid.
Remark 2.2. If I is an ideal of g, we have: I absolutely Aut-invariant ⇒ I Aut(g)-invariant ⇒ I Aut(g)0-invariant (H0 denoting the connected component of the unit in the Zariski topology). The reverse implications do not hold in general. For instance, insl2(K)×sl2(K), the idealsl2(K)× {0}only satisfies the third condition. Also, over the reals, in sl2(R)×so3(R), the ideal sl2(R)× {0}
only satisfies the latter two conditions (however, it is solid). Over the complex numbers, I do not know whether every solid ideal is Aut(g)-invariant.
Theorem 2.1 follows from the next two propositions. Since the Lie algebras axioms plays no role here, we consider arbitrary algebras and the context could be even more general. Also, solid can be defined for arbitrary subspaces and we use this straightforward generalization. However, in a Lie algebra solid subspaces are always ideals (Corollary 2.6)
Proposition 2.3. Let g be a fd K-algebra and I a K-subspace, Q-defined for at least one multiplicative Q-structure; let K¯ be an algebraically closed extension of K. If I ⊗K K¯ is Aut(g)K¯-invariant and Q-defined for some multiplicative Q-structure on the algebra g, then I is solid.
Proof. Let Ξ be the Galois group of ¯K overQ.
Let V be a finite-dimensional K-vector space. Let W ⊂ V be a Q-structure in V. Then Ξ acts coordinate-wise on VK¯ = V ⊗K K¯ = W ⊗Q K¯ (for some choice of basis of W, whose choice does not matter). We denote this action as γ ·v = uW(γ)v for γ ∈ Ξ. Then uW(γ) is a Q-linear automorphism of VK¯; it is also γ-semi-linear, in the sense that uW(γ)(λv) = γ(λ)uW(γ)v. It follows in particular that if W0 is another Q-structure, then ηW,W0(γ) =u−1W(γ)uW0(γ) is a K-linear automorphism of¯ VK¯.
Now suppose that the actionuW of Ξ leaves a ¯K-subspace J ⊂ VK¯ invariant.
This implies thatJ is the ¯K-linear span ofJ∩W (see [Bou2, corollairein Chap.
V.4]).
We apply this toJ = IK¯ =I ⊗KK¯. By assumption, I is Q-defined for some multiplicative Q-structure W0. So J is uW0(Ξ)-invariant. It is also Aut(g)( ¯K)- invariant, by assumption. Hence, by the first paragraph of the proof, it isuW(Ξ)- invariant. SoIK¯ is the ¯K-linear span of J∩W. This means that dimQ(J∩W) = dimK¯(IK¯). Since the latter equals dimK(I) and since J ∩W = I ∩W, this in turn implies thatI is the K-linear span of I∩W, that is, I isQ-defined.
Proposition 2.4. Letgbe a fd algebra over an uncountable field of characteristic zero, definable overQ. LetV be a subspace ofg. Suppose thatV is solid, that is, it isQ-defined for everyQ-structure ong. ThenV is invariant underH = Aut(g)0, that is, when K has characteristic zero, V is stable under every derivation of g.
Proof. LetL⊂Hbe the stabilizer ofV. IfL6=H, thenH(K)/L(K) is uncount- able: if K is R or C this is because it is a manifold of positive dimension; in general see Lemma 2.5. The map hL→hV fromH(K)/L(K) to the set of sub- spaces of g being injective, it has an uncountable image. So, given Q-structure gQ, there exists h ∈ H(K) such that hV is not Q-defined. Accordingly, for the new Q-structure defined by h−1gQ, V is not Q-defined, contradicting that V is
solid.
Lemma 2.5. Let H be a connected linear algebraic group defined over an infinite perfect field K, and L a K-closed proper subgroup. Then H(K)/L(K)is has the same cardinality as K.
Proof. (Beware that the canonical injective map H(K)/L(K)→(H/L)(K) can fail to be surjective, so it is not enough to compute the cardinal of the latter.)
Clearly the cardinal ofH(K) is bounded above by that of K (as soon asK is infinite), so we have to prove the other inequality.
We first take for granted that there exists a K-closed curve C in H, K- birational to P1, such that C * L and 1 ∈ C. This granted, let us conclude (without the perfectness restriction onK).
OnC, we consider the equivalence relation x∼yif x−1y∈L. This is a closed subvariety ofC×C, and does not contain any layerC×{y0}sincey0 ∈Land then C⊂L would follow. So equivalence classes are finite. SinceC isK-birational to P1, its cardinal is the same asK, and then its image in the quotientH(K)/L(K) being the quotient of C(K) by the equivalence relation ∼ with finite classes, it also has at least the cardinal ofK.
To justify the existence of C, we can argue that H is K-unirational (this uses that K is perfect), as established in [BorSpr, Corollary 7.12]; consider a dominantK-defined morphismf :U →H, withU open in the affined-spaceAd. Conjugating with translations both inAd andH, we can suppose that 0∈U and f(0) = 1. Since U(K) is Zariski-dense and f is dominant, f(U) is Zariski-dense,
and hence it contains a point f(x) ∈/ L for some x∈U. Let D⊂ Ad be the line through x. Then the Zariski closure off(D∩U) is the desired curve.
Corollary 2.6. In a fd Lie algebra over a field of characteristic 0, every solid subspace is an ideal.
Proof. We have to show that ifV is Aut(g)0-invariant then it is an ideal. Indeed, differentiation implies thatV is invariant under derivations ofg, and this includes inner derivations. Since left and right multiplications are derivations, this finishes
the proof.
Example 2.7 (A solid complete flag in a filiform Lie algebra of dimension ≥ 4).
Recall that, for d ≥ 2, a filiform Lie algebra denotes a d-dimensional nilpotent Lie algebra whose nilpotency length is exactlyd−1, which is the largest possible value. Such a Lie algebra gadmits a basis (e1, . . . , ed) such that, denoting byg≥i
the subspace with basis (ei, . . . , ed), we have, for alli∈ {2, . . . , d},gi =g≥i+1. In addition, if d ≥ 4, we can arrange to choose [e1, e2] = e3 and [e1, e3] = e4. This being assumed, each g≥i is a solid ideal. Indeed, for i6= 2, this is because it is a term of the lower central series. For i= 2, this is because it is the centralizer of g2(= g≥3) modulog4(=g≥5).
3. Systolic growths: facts and bounds
3.1. Asymptotic comparison. Given functions f, g (of a positive real variable r), we writef g if f(r)≤Cg(C0r) +C00 for some constants C, C0, C00 >0 and all r. We say that f and g are '-equivalent and write f 'g if f g f. Also, we write f g if g/(|f|+ 1) → +∞ (usually limr→∞f > 0, in which case this just means g/f →+∞).
3.2. Residual girth. The residual girth is defined in the same way as the systolic growth, but allowing only normal subgroups.
LetGbe a simply connected nilpotent Lie group, of dimensiondand nilpotency length c with a lattice Γ. We can find an embedding of G into the group of upper triangular unipotent real matrices, mapping Γ into integral matrices. Then congruence subgroups (the kernel of reduction modulon, in restriction to Γ) have index 'ndand systole 'n1/c. This yields for Γ the polynomial upper bound on the residual girthr 7→rcd; this simple observation was made independently in [BouStu, Cor]. As observed in [Cor], in the case of the 3-dimensional Heisenberg group, it is sharp: the residual girth of every lattice is ' r6. In [BouStu], it is shown that, more generally, the residual girth of every lattice is indeed ' rcd when the center ofGcoincides with thec-th term of the central series. Otherwise the picture is not completely clear.
The above provides an easy polynomial upper bound on the systolic growth of Γ and of G, which are also rcd. Nevertheless, we will not concentrate further on the residual girth σCΓ, inasmuch as its asymptotic behavior is usually much
larger than the systolic growthσΓ: as soon asGis non-abelian, σΓ(r)rcd, and it is also very likely that σΓ σΓC always holds.
3.3. Systolic growths. Another notion, also related with conjugacy phenom- ena, but more closely related to the systolic growth, is the uniform systolic growth introduced in [Cor]: we define it now.
Let H be an arbitrary compactly generated locally compact group. We use the standard natural convention inf∅ = +∞. In the setting we will study more deeply (compactly generated locally compact nilpotent groups), all lattices are cocompact. So we stick to cocompact lattices, but in general it would make sense to consider the analogous notion allowing arbitrary lattices.
In the setting of Definition 1.1, one can define the uniform systole, orH-uniform systole, of X ⊂H as the infimum
h∈Hinf sys(hXh−1) = inf
|hxh−1|: h∈H, x∈Xr{1} .
The uniform (orH-uniform) systolic growth ofHis then defined as the function mapping r > 0 to the infimum σu(r)∈ R+∪ {+∞} of covolumes of cocompact lattices ofH with uniform systole ≥r.
Given a cocompact lattice Γ inH, we can consider the uniform systolic growth σΓof Γ (computed within Γ) and the uniform systolic growthσH ofH. But while σH σΓ, a similar estimate for the uniform systolic growths might fail because for a finite index subgroup of Γ, theH-uniform systole can be much smaller than the Γ-uniform systole (see Example 3.2, in the Heisenberg group). Hence another notion naturally appears: the H-uniform systolic growth σΓ,Hu of Γ, considering the uniform systole computed in H but finite index subgroups of Γ. At this point we have a bunch of growths, which we summarize now (up to asymptotic equivalence, allowing to not specify the choice of lengths): each maps r ≥ 0 to the smallest covolume of a cocompact lattice ofH with the additional conditions:
• (systolic growth σH of H): of systole ≥r;
• (systolic growth σΓ of Γ: contained in Γ, of systole≥r;
• (uniform systolic growth σHu of H): ofH-uniform systole ≥r
• (uniform systolic growth σuΓ of Γ): contained in Γ, of Γ-uniform systole
≥r
• (H-uniform systolic growth σuΓ,H of Γ): contained in Γ, of H-uniform systole ≥r.
with asymptotic inequalities
σH σΓ σuΓσΓ,Hu ; σH σHu σΓ,Hu .
(Let us also mention that all these functions areσCΓ: the only nontrivial case is that of σΓ,Hu , and follows from the fact that for a subgroup Λ normalized by a fixed cocompact lattice Γ, the H-normal systole is bounded by the Γ-uniform systole plus a constant depending only on Γ, independently of Λ.)
In the context when H = G is a simply connected nilpotent Lie group, all these functions take finite values and are asymptotically bounded above by the residual girth of Γ, and in particular are polynomially bounded by r7→rcdim(G), a better upper bound (implying rcdim(G)/2) is provided in Proposition 5.1. In this context, we do not know if all these five functions have the same asymptotic behavior (Question 1.10). In various cases where we prove an upper bound on the systolic growth, by constructing an explicit sequence of lattices, we actually provide an upper bound on σuΓ,G and therefore on all others.
Remark 3.1. Most of these functions can be interpreted in the geometry ofG/Γ.
More precisely, endow G with a right-invariant Riemannian metric, which thus passes to the quotient G/Γ, as well as its covering G/Λ when Λ is a finite index subgroup of Γ. Then Γ can naturally be identified to π1(G/Γ) by a bijection γ 7→ jγ (the base-point is meant to be the obvious one). The length of γ ∈ Γ is equivalent to the length in X = G/Γ of a smallest representative based loop of jγ. Its G-uniform length is equivalent to the infimum of lengths of arbitrary loops in the free homotopy class of jγ. Therefore, the systole (resp. G-uniform systole) of Λ is equivalent the smallest size of a based loop (resp. of a loop) in X not homotopic to a point. Call the latter the geometric based systole, resp.
geometric systole, of X (classically, the word “geometric” is dropped, since these notions come from Riemannian geometry!). The geometric based systolic growth, resp. geometric systolic growth, ofX is defined as the function mappingr to the smallest degree of a covering of X with geometric based systole, resp. systole
≥ r. Thus the geometric based systolic growth of X coincides with the systolic growth of Γ, and the geometric systolic growth ofX coincides with theG-uniform systolic growth of Γ.
Example 3.2. In the 3-dimensional real Heisenberg groupG, let us write, for the sake of shortness,M(a, b, c) =
1 a c 0 1 b 0 0 1
. We write|M(a, b, c)|=|a|+|b|+p
|c|;
we use this approximation of the distance to the origin to compute systoles.
Let Γn be the subgroup generated by the matrices xn = M(1,0, n), yn = M(0, n2,0); this is a lattice. We claim that its G-uniform systole is 1 while its Γn-uniform systole is≥√
n.
Let us describe Γn. Define zn = xnynx−1n yn−1. Then zn = M(0,0, n2) and elements of Γn are precisely those xanybnznc when (a, b, c) ranges over Z3. We see that
xanynbzcn=M(a, bn2, n(a+ (ab+c)n)).
In G, xn is conjugate to M(1,0, t) for every real t, and hence the G-systole of Γn is equal to 1.
Let us compute the Γn-uniform systole. Consider (a, b, c) ∈ Z3 r {0} and considerg =xanybnznc, andg0 any conjugate ofg (for the moment, a G-conjugate).
• If (a, b) = (0,0), thenc6= 0 andg =M(0,0, cn2) is central, so|g0|=|g|= p|c|n≥n;
• if b 6= 0, then |g0| ≥ |a|+|b|n2 ≥n2;
• if b = 0 anda6= 0, then g =xanznc =M(a,0, n(a+cn)). We discuss – if |a| ≥√
n, then|g0| ≥√ n;
– if |a| < √
n, we now assume that g0 is a Γn-conjugate of g, namely by some element M(∗, dn2,∗0) with d ∈ Z (we do not have to care about the coefficients denoted by stars); this gives
g0 =M(a,0, n(a−(ad−c)n)).
If by contradiction |g0| < √
n, then a −(ad −c)n = 0, but since
|a|< n this implies a= 0, a contradiction. So |g0| ≥√ n.
We conclude that in all cases|g0| ≥√
n, so the Γn-systole of Γn is ≥√ n.
4. Algebraization of the systolic growth
The purpose of this section is to describe the various systolic notions in terms of discrete cocompact subrings of the Lie algebra, instead of lattices in the Lie group. While the exponential of a discrete cocompact subring can fail to be a subgroup and the logarithm of a lattice can fail to be an additive subgroup or fail to be stable under taking brackets, the correspondence is true up to bounded index. This is the contents of the following lemma.
Lemma 4.1 (Folklore). Let g be a real fd nilpotent Lie algebra and G the cor- responding simply connected nilpotent Lie group. There exists C ≥ 1 depending only on dim(g) such that:
(1) for every cocompact discrete subring Λ in g, there exist lattices Γ1,Γ2 in G with Γ1 ⊂exp(Λ)⊂Γ2 and [Γ2 : Γ1]≤C.
(2) for every lattice Γ in G, there exist cocompact discrete subrings Λ1,Λ2 in g with Λ1 ⊂log(Γ)⊂Λ2 and [Λ2 : Λ1]≤C.
We include a proof (of a slightly stronger statement) in Appendix A.
4.1. Systolic growth of a real nilpotent Lie algebra. We define a Lie algebra analogue of the systole and systolic growth as follows (fix some Lebesgue measure on the vector spaceg). Recall that b·c denotes the Guivarch length, introduced in§2.1.
Definition 4.2. IfH ⊂g, define sys(H) = inf
bhc:h∈Hr{0} ∈R+∪ {+∞}.
Also define the uniform (or G-uniform) systole as sysu(H) = infg∈Gsys(g ·H) (whereG acts through the adjoint representation).
The systolic (resp. uniform systolic) growth of (g,b·c) is the function mapping r ≥ 0 to the infimum σ(r) ∈ R+ ∪ {+∞} of covolumes of cocompact discrete subrings ofg with systole (resp. uniform systole) ≥r.
Note that the systolic growth also depends on the choice of normalization of the Lebesgue measure. Its asymptotic growth, however, only depends on the real Lie algebra g. Its interest lies in the following fact:
Proposition 4.3. The systolic (resp. uniform systolic) growth of g (as a Lie algebra) and the systolic (resp. uniform systolic) growth of G are '-equivalent.
Proof. The exponential map preserves the systole of subsets up to a bounded mul- tiplicative error, by Guivarch’s estimates. So the proposition would be already proved if the exponential map were giving an exact correspondence between co- compact discrete subrings of g and lattices in G. This is not the case, but is, however, “true up to a bounded index error”, as explained in Lemma 4.1, which entails the result for systolic growth.
The uniform case follows, using in addition that the exponential mapg→Gis G-equivariant, for the adjoint action on g and the conjugation action onG.
4.2. Systolic growth of a rational nilpotent Lie algebra. To estimate the systolic growth in the lattice, we need a similar notion pertaining to rational Lie algebras. Namely, let l be a fd nilpotent Lie algebra over Q. Define its
“realification” g =l⊗QR. Choose vi and norms as above on g, so as to define the Guivarch length b·c; so we have a notion of systole for subsets g, and choose a Lebesgue measure on g.
Definition 4.4. The systolic (resp.G-uniform systolic) growth of the rational Lie algebra (l,b·c) is the function mapping r≥0 to the infimum σ(r)∈R+∪ {+∞}
of covolumes of cocompact discrete subrings ofgcontained inl, with systole (resp.
G-uniform systole) ≥r.
Note that the only difference in the last definition is that we restrict to those subrings contained in l. In particular, this function is bounded below by the systolic growth of G (relative to the same choice of norms, etc.). Also note that we did not attempt to define an analogue of the Γ-uniform systolic growth.
Denoting L= exp(l) (which can be thought as the groupGQ of Q-points of G for a suitable rational structure), we have
Proposition 4.5. IfΓ is any lattice inGcontained in L, then the systolic (resp.
G-uniform systolic) growth of Γ is '-equivalent to the systolic (resp. G-uniform systolic) growth of the rational Lie algebra l.
Proof. An observation is that in Lemma 4.1, if Λ⊂l then automatically Γi ⊂L, and in the other direction if Γ⊂Lthen automatically Λi ⊂l. This being granted, the proof follows the same lines as that of Proposition 4.3.
5. General upper bounds on the systolic growth
We prove here Proposition 1.3, giving here a more general result since we consider the uniform systolic growth.
Proposition 5.1. Let Gbe a simply connected nilpotent Lie group with a lattice Γ. Let D be its homogeneous dimension and let k be defined in §1.3. Then σuΓ,G(r) rD+k, and hence the systolic growth and G-uniform systolic growth of both G and Γ are all rD+k.
Proof. It is enough to show that the G-uniform systolic growth of Γ is rk0, where
k0 = c
2dim(g/gdc/2e) +
c
X
i=dc/2e
idim(gi/gi+1) =
c
X
1
max(c/2, i) dim(gi/gi+1).
Let GQ ⊂ G be the rational Malcev closure of Γ and gQ its Lie algebra. From the lower central series we choose supplement subspace to obtain a vector space decomposition g = Lc
i=1gi, with L
j≥igj = gi for all i (this is not necessarily a Lie algebra grading). We choose bases of these subspaces so as to ensure that all structure constants are integral; thusgi(Z) means the discrete subgroup generated by the given basis ofgi.
We now define, for every square integern, Λn=Lc
i=1nmax(c/2,i)gi(Z). We have to check that this is a subalgebra, namely that
Bij := [nmax(c/2,i)gi(Z), nmax(c/2,j)gj(Z)]⊂Λn
for all i, j. Indeed, keeping in mind thatn is a square, Bij ⊂nc[gi(Z),gj(Z)]⊂nc
c
M
p=i+j
gp(Z)⊂Λn.
The G-uniform systole of Λn is n: indeed, any nonzero element has the form w = P
j≥injvj with vi ∈ gi(Z)r{0}. So any G-conjugate of w has the form nivi+µ with µ ∈ gi+1 and hence has norm ≥ ni (for the `1-norm with respect to the fixed basis), and thus has Guivarch length ≥ n. Then Λn precisely has covolumenk0.
Using that the exponential mapg→GisG-equivariant and in view of Lemma
4.1, we deduce the desired upper bound.
The following proposition provides upper bounds onk andk0 =k+Ddepend- ing only on the dimension d. Recall that the maximal homogeneous dimension D for given dimension d ≥ 2 is equal to d(d−1)2 + 1 and is precisely attained for filiform Lie algebras, which are those of maximal nilpotency class (namelyd−1).
Proposition 5.2. Letgbe a finite-dimensional nilpotent Lie algebra of dimension d, nilpotency length c, and homogeneous dimension D, and k =kc(g) is defined in §1.3. Then
k≤ d2 6 − d
2 +1
2 and k+D≤ 5d2 −4d
8 .
Proof. We denote b=dc/2e and d= dim(g). Then k =
b−1
X
i=1
c 2−i
dim(gi/gi+1)
=c
2dim(g/gb)−dim(g/g2)−
b−1
X
i=2
idim(gi/gi+1);
=c
2(d−dim(gb))−(d−dim(g2))−
b−1
X
i=2
idim(gi/gi+1);
fori≥2 we use the inequality dim(gi/gi+1)≥1, dim(gi)≥c−i+ 1 for 1≤i≤c (applied for i= 2 and i=b) and get
k ≤c
2(d−c+b−1)−(d−c+ 1)− b(b−1) 2 + 1;
write c= 2b−e with e ∈ {0,1}, this yields
k ≤ −3c2+ 2(2d+ 3)c−8d+e
8 .
A polynomial function of the form −3x2+ 2ax is maximal for x=a/3, where it takes the value a2/3. Hence
k ≤
1
3(2d+ 3)2−8d+e
8 = 1
6d2− 1
2d+ 3 +e 8 .
Now consider k0 =k+D and again write c= 2b−e withe ∈ {0,1}. Then k0 =c
2dimg+
c
X
i=b
i− c
2
dim(gi/gi+1)
=cd 2 +
c
X
i=b
i− c
2
+
c
X
i=b
i− c
2
(dim(gi/gi+1)−1);
≤cd 2 +
c
X
i=b
i− c
2
+ c 2
c
X
i=b
(dim(gi/gi+1)−1);
= cd 2 +
c
X
i=b
(i−c) + c 2
c
X
i=b
dim(gi/gi+1);
= cd
2 + c(c+ 1)
2 −b(b−1)
2 −c(c−b+ 1) + c 2
c
X
i=b
(dim(gi/gi+1)−1);
= 4cd−c2+ (2e−2)c+e
8 + c
2dim(gb).
For all i ≥ 2 we have dimgi ≤ d−i. Therefore we have (assuming c≥ 3, so b≥2)
k0 ≤ 4cd−c2+ (2e−2)c+e
8 + c
2(d−b) = 8cd−3c2−2c+e
8 .
A function of the form x 7→ −3x2 +ax increases until it takes its maximum for x=a/6; actually, using d ≥4,d−1≤ (8d−7e−6)/6, and since c≤d−1, so the last expression is bounded above by the same whenc is replaced with its maximum possible value d−1. Hence
k0 ≤ 8(d−1)d−3(d−1)2−2(d−1) +e
8 = 5d2−4d+e−1
8 .
6. The strategy for lower bounds
We consider a fd c-step nilpotent real Lie algebra g, with lower central series (gi)i≥1 and dimension d. We decompose it as a direct sum of subspaces g = Lc
i=1vi, so that for alli,gi =L
j≥ivj (we call this a compatible decomposition).
Let us choose a basis of g as a concatenation of bases of the vi. Note that the basis determines the subspaces vi, which is spanned by the subset of basis elements that belong togirgi+1.
It is convenient to also directly define such a basis (without defining the vi beforehand). A basis (e1, . . . , ed) in a fd nilpotent Lie algebra is compatible if it satisfies the following three conditions:
• for all i, j, (ej ∈girgi+1 and ek ∈gi+1) implies j < k;
• {ej :j ≥1} ∩gi spans gi for all i;
• the subspaces gi =hej :j ≥1, ej ∈gi rgi+1i span their direct sum.
Thus any compatible basis determines a compatible decomposition, and any compatible decomposition yields compatible bases.
In the real case, it is convenient to assume that the nonzero structure constants ofg with respect to this basis have absolute value≥1 (this is a mild assumption as it always hold after replacing the basis by a scalar multiple, thus not changing the compatible decomposition).
We define a compatible flag as a sequence of ideals (6.1) g=w1 >w2 >· · ·>wk ={0}, such that each term gi occurs among the wj.
6.1. Dilations. Fix a nilpotent fd Lie algebra with a compatible decomposition g=Lc
i=1vi.
For a nonzero scalarrdefine the diagonal linear automorphism u(r) of ggiven by multiplication byri on vi.
We define a new Lie algebra g[r], with underlying k-linear space g, with the bracket [x, y]r =u(r)−1[u(r)x, u(r)y], that is, the pull-back of the original bracket by the linear automorphismu(r).
Example 6.1. Consider the 5-dimensional Lie algebra with basis (e1, . . . , e5) and nonzero brackets [e1, e3] = e4, [e1, e4] = [e2, e3] = e5. This is a compatible basis, withv1,v2,v3having bases (e1, e2, e3), (e4) and (e5) respectively. Theng[r] has, in this basis, the nonzero brackets [e1, e3]r =e4, [e1, e4]r=e5, and [e2, e3]r =r−1e5. Remark 6.2. If g = L
vi is a Lie algebra grading (and thus a Carnot grading), then [·,·]r = [·,·]. Beware that for an arbitrary (e.g., Carnot) nilpotent Lie algebra of nilpotency length c ≥ 3, we can find a compatible decomposition g = L
vi such that [v1,v1] is not contained in v2.
Remark 6.3. We have g=g[1], and the function (g, h, r)7→[g, h]r is polynomial with respect tor−1 (with each coefficient a skew-symmetric bilinear mapg×g→ g). The constant coefficient [·,·]∞ of this polynomial can be thought as the limit of the brackets [·,·]r when r → ∞ (this is indeed the case when K is a normed field), and (g,[·,·]∞) is the associated Carnot-graded Lie algebra of g, in which for x∈vi and y∈vj, the bracket [x, y]∞ is defined as the projection of [x, y] on vi+j modulo gi+j+1. This 1-parameter family of brackets has been used by Pansu and Breuillard [Pan, Bre].
6.2. Renormalization of the algebraic systolic growth. As in §6.1, we fix a nilpotent fd Lie algebra with a compatible decomposition g = Lc
i=1vi, and assume in addition that the ground field is the field of real numbers. Then the dilation u(r) multiplies the Lebesgue measure (for any choice of normalization) by |r|D, where D is the homogeneous dimension of g (see §2.1).
We fix a norm on gwhich is the sup-norm with respect to this decomposition, that is, for every x =P
xi, xi ∈vi, we have kxk = maxkxik. This implies that u(r) multiplies the Guivarch length by |r|. This also implies that for x ∈g, the conditions kxk ≥1 and bxc ≥1 are equivalent.
Let Λr be a discrete cocompact subring of Guivarch systole≥r. For instance, assuming that gadmits a rational structure, we can choose, by compactness, Λr to have covolume exactly σ(r) (i.e., has minimal covolume among those discrete cocompact subrings of Guivarch systole ≥r.
Then Ξr = u(r)−1Λr has Guivarch systole ≥ 1; this is a discrete cocompact subring in g[r]. Its covolume satisfies cov(Ξr) = r−Dcov(Λr).
The idea is that rather than studying Λr (whose systole tends to infinity), we study Ξr (which has bounded systole≥1), the only caveat being that Ξr is a Lie subring of g[r] (i.e., for some Lie bracket which varies with r).
6.3. First application: small systolic growth. Assume here that limr−Dσ(r)<
∞ and let us prove thatg is Carnot.
Suppose that cov(Λr) ≤ CrD (for r ∈ I, where I is an unbounded set of positive real numbers). Then cov(Ξr)≤C. By compactness of the set of lattices with systole ≥ 1 and covolume ≤ C, we can find an unbounded subset J ⊂ I such that limJ3r→∞Ξr = Ξ∞ (in the Chabauty topology) for some lattice Ξ∞. Clearly Ξ∞ is a subring of (g,[·,·]∞).
Then Ξr is isomorphic as a Lie ring to Ξ∞ for large enough r ∈ J. Indeed, choose a basis (u1, . . . , ud) of Ξ∞ as a Z-module. Then we can find uri ∈Ξr with uri → ui. Then [ui, uj] = P
knkijuk for suitable integers nkij (here exponents are additional indices, and not powers). Then [uri, urj]−P
knkijurk belongs to Ξr and converges to zero, hence is zero forr large enough. Moreover, since the covolume is a continuous function, eventually (uri) is a Z-basis of Ξr. This shows that this is indeed an isomorphism. Sinceg'Ξr⊗ZR and (g,[·,·]∞)'Ξ∞⊗ZR as real Lie algebras, this shows that (g,[·,·]∞)'g. Thus g is Carnot.
6.4. Covolume inequalities. (In this §6.4, the Lie algebra bracket plays no role.) Assume now that the ground field is the field of real numbers.
We fix a compatible flag as in (6.1), denoted F for short. We fix Lebesgue measures on all wj/wj−1, and thus on g. Fix a compatible basis, compatible with this flag. This basis defines (in a compatible way) normalizations of the Lebesgue measures on each quotientwj/wk, where the cube [0,1[q (for the given basis) has volume 1, and sup norms with respect to this basis.
The basis also yields a compatible decomposition; we thus have a notion of Guivarch length (§2.1). Observe that u(r) multiplies the Guivarch length by|r|.
We say that an additive lattice Λ of (g,+) isF-compatible if Λ∩wj is a lattice in wj for every j. This implies that for all j the projection Λ[j] of Λ∩wj on fj = wj/wj−1 is a lattice in fj as well. Let aj(Λ) be the covolume of Λ[j] in fj. Then the covolume of Λ∩wj is equal toQ
k≥jak(Λ).
Note that for any v ∈ g, we have kvk ≥ 1 if and only if bvc ≥ 1. So for an additive lattice of systole ≥ 1 (for either the norm or the Guivarch length), the covolume is≥ 1. Thus,Q
k≥jak(Λ) ≥1 for all j. This means that the following holds:
Lemma 6.4. For anyj, anyF-compatible additive lattice of gof Guivarch systole
≥1 satisfies Y
k≥j
ak(Λ)≥1, or equivalently, X
k≥j
logr(ak(Λ))≥0, ∀r >1.
In the sequel, these inequalities will be combined with other inequalities making use of further assumptions (namely that Λ is a Lie subring).
The requirements about the norm will be fulfilled when we consider a Lie algebra with basis (e1, . . . , ed), and a flag containing all elements of the lower central series such that, denoting g≥i, each element of the flag is one of the g≥i; on each g≥i/g≥j the norm being the `∞ norm and the Lebesgue measure being normalized so that the cube [0,1[k has measure 1.
7. Precise estimates
We give explicit estimates of the systolic growth for various illustrating exam- ples. While we obtain upper bounds by easy explicit construction, we use the method of§6 to obtain lower bounds.
7.1. 5-dimensional non-Carnot Lie algebras. Following a convenient cus- tom, when we describe a Lie algebra by saying that the nonzero brackets are [ei, ej] = fij, we mean that [ej, ei] = −fij and that all other brackets between basis elements are zero. That an algebra defined in such a way is indeed Lie is equivalent to say that
J(ei, ej, ek) = [ei,[ej, ek]] + [ej,[ek, ei]] + [ek,[ei, ej]] = 0, ∀i < j < k.
There are exactly two non-isomorphic non-Carnot nilpotent 5-dimensional real Lie algebras. Using the notation in [Gra], these are defined, in the basis (e1, . . . , e5), by the nonzero brackets:
L5,5 : [e1, e2] =e4, [e1, e4] =e5, [e2, e3] =e5; L5,6 : [e1, e2] =e3, [e1, e3] =e4, [e1, e4] =e5, [e2, e3] =e5. The lower central series in L5,5 is 123/4/5 (this concise notation meansg2 =g≥4 and g3 =g≥5) and in L5,6 it is 12/3/4/5. Thus we can write the Lie algebra law of g[r] in each case:
L5,5 : [e1, e2]r =e4, [e1, e4]r =e5, [e2, e3]r =r−1e5; L5,6 : [e1, e2]r =e3, [e1, e3]r=e4, [e1, e4]r =e5, [e2, e3]r =r−1e5. Lemma 7.1. In both cases, the complete flag(g≥i)1≤i≤5 is made up of solid ideals.
Proof. All are part of the lower central series, except g≥2 in both cases and g≥3
for L5,5. The upper central series is 12/34/5 for L5,5, so g≥3 is also solid in this case. Finally, g≥2 is the centralizer of g≥4 in both cases, so is solid.
Let now, in either case, Λr be a discrete cocompact subring ing, with Guivarch systole ≥ r and covolume σ(r), and define Ξr = u(r)−1Λr, which is a discrete cocompact subring of g[r] with systole ≥1. As in§6.4, let ai(r) be the systole of the projection of Ξr∩gi in gi/gi+1, and write Ai = logr(ai(r)).
Lemma 7.2. In both cases, we have, for all r >0, the inequalitiesA1+A4 ≥0, A2+A3 ≥1, A5 ≥0. In particular, P5
i=1Ai ≥1.
Proof. It is convenient to denote by o(i), resp. O(i), an unspecified element of g≥i+1, resp. g≥i. By definition, Ξr contains elements v1 = a1e1 +o(1), v2 = a2e2+o(2), v3 =a3e3+o(3), v4 =a4e4+o(4), v5 =a5e5.
Then A5 ≥ 0 means a5 = kv5k ≥ 1, which is a trivial consequence of having systole ≥1.
Then [o(1), O(4)] = [O(1), o(4)] = [o(2), O(3)] = [O(2), o(3)] = 0 in both cases.
Since for both L5,5[r] and L5,6[r] we have [e1, e4]r = e5 and [e2, e3]r = r−1e5, it follows that [v1, v4]r = a1a4e5 and [v2, v3]r =r−1a2a3e5. Therefore a1a4 ≥1 and r−1a2a3 ≥1, which means thatA1+A4 ≥0 andA2+A3 ≥1. The last inequality follows
A1 +A2+A3+A4+A5 = (A1+A4) + (A2+A3) +A5 ≥0 + 1 + 0 = 1.
